IMHO your problem is more likely related to an unlikely problem with ORACLE_HOME, ORACLE_SID setup or to a unlikely bug that would occur when SQL*Plus tries to check that the ORACLE_SID instance is running.

Can you try to trace SQL*Plus with:

strace -fo /tmp/sqlplus.strace sqlplus '/ as sysdba'

Your problem has some similarity with the following one (only for 10.2.0.1 and RAC):

It seems you have a mismatch between the Unix group of the Unix account that has started the database instance (sapsys) and the Unix group of the Oracle software owner (dba).
Is bp3adm also member of the Unix "dba" group ?

If you would change bp3adm group to "dba" or if Oracle software would be owned by bp3adm account, you should not have these connection problems.
But these installation changes that would require carefull testing. See http://download-uk.oracle.com/docs/cd/B19306_01/server.102/b15658/admin_ora.htm#sthref115

My guess is that you need to allow set user and group id on the oracle software mount point and restart the instance.

The trace command has generated a huge file. I am
attaching a part of the trace file.

That part does not tell me much, no more than loading library files. Output right before sqlplus outputs "connected to idle instance" is perhaps more useful but on the other hand I suspect analyzing the output is hard to do over the forum.

For shm system calls, you have some system calls that look OK and some that fail. Difficult to say whether this is OK or not: this is something more for Oracle support.

I would also recommend you what orafad is telling you. And to carefully review how you want to install your system from Oracle point of view using Oracle and SAP documentation:
- which account/group must be the owner of the Oracle software
- which account/group must be the owner of the Oracle processes = database instance
Oracle requires at least that these 2 accounts must belong to the same group (generic name is OSDBA in Oracle documentation).